\(\int x^2 (a+b x^2)^2 (c+d x^2) \, dx\) [143]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \]

[Out]

1/3*a^2*c*x^3+1/5*a*(a*d+2*b*c)*x^5+1/7*b*(2*a*d+b*c)*x^7+1/9*b^2*d*x^9

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{7} b x^7 (2 a d+b c)+\frac {1}{5} a x^5 (a d+2 b c)+\frac {1}{9} b^2 d x^9 \]

[In]

Int[x^2*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(a^2*c*x^3)/3 + (a*(2*b*c + a*d)*x^5)/5 + (b*(b*c + 2*a*d)*x^7)/7 + (b^2*d*x^9)/9

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c x^2+a (2 b c+a d) x^4+b (b c+2 a d) x^6+b^2 d x^8\right ) \, dx \\ & = \frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \]

[In]

Integrate[x^2*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(a^2*c*x^3)/3 + (a*(2*b*c + a*d)*x^5)/5 + (b*(b*c + 2*a*d)*x^7)/7 + (b^2*d*x^9)/9

Maple [A] (verified)

Time = 2.62 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b^{2} d \,x^{9}}{9}+\frac {\left (2 a b d +b^{2} c \right ) x^{7}}{7}+\frac {\left (a^{2} d +2 a b c \right ) x^{5}}{5}+\frac {a^{2} c \,x^{3}}{3}\)	\(52\)
norman	\(\frac {b^{2} d \,x^{9}}{9}+\left (\frac {2}{7} a b d +\frac {1}{7} b^{2} c \right ) x^{7}+\left (\frac {1}{5} a^{2} d +\frac {2}{5} a b c \right ) x^{5}+\frac {a^{2} c \,x^{3}}{3}\)	\(52\)
gosper	\(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\)	\(54\)
risch	\(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\)	\(54\)
parallelrisch	\(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\)	\(54\)

[In]

int(x^2*(b*x^2+a)^2*(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

1/9*b^2*d*x^9+1/7*(2*a*b*d+b^2*c)*x^7+1/5*(a^2*d+2*a*b*c)*x^5+1/3*a^2*c*x^3

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac {1}{3} \, a^{2} c x^{3} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{5} \]

[In]

integrate(x^2*(b*x^2+a)^2*(d*x^2+c),x, algorithm="fricas")

[Out]

1/9*b^2*d*x^9 + 1/7*(b^2*c + 2*a*b*d)*x^7 + 1/3*a^2*c*x^3 + 1/5*(2*a*b*c + a^2*d)*x^5

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {a^{2} c x^{3}}{3} + \frac {b^{2} d x^{9}}{9} + x^{7} \cdot \left (\frac {2 a b d}{7} + \frac {b^{2} c}{7}\right ) + x^{5} \left (\frac {a^{2} d}{5} + \frac {2 a b c}{5}\right ) \]

[In]

integrate(x**2*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

a**2*c*x**3/3 + b**2*d*x**9/9 + x**7*(2*a*b*d/7 + b**2*c/7) + x**5*(a**2*d/5 + 2*a*b*c/5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac {1}{3} \, a^{2} c x^{3} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{5} \]

[In]

integrate(x^2*(b*x^2+a)^2*(d*x^2+c),x, algorithm="maxima")

[Out]

1/9*b^2*d*x^9 + 1/7*(b^2*c + 2*a*b*d)*x^7 + 1/3*a^2*c*x^3 + 1/5*(2*a*b*c + a^2*d)*x^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, b^{2} c x^{7} + \frac {2}{7} \, a b d x^{7} + \frac {2}{5} \, a b c x^{5} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{3} \, a^{2} c x^{3} \]

[In]

integrate(x^2*(b*x^2+a)^2*(d*x^2+c),x, algorithm="giac")

[Out]

1/9*b^2*d*x^9 + 1/7*b^2*c*x^7 + 2/7*a*b*d*x^7 + 2/5*a*b*c*x^5 + 1/5*a^2*d*x^5 + 1/3*a^2*c*x^3

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^5\,\left (\frac {d\,a^2}{5}+\frac {2\,b\,c\,a}{5}\right )+x^7\,\left (\frac {c\,b^2}{7}+\frac {2\,a\,d\,b}{7}\right )+\frac {a^2\,c\,x^3}{3}+\frac {b^2\,d\,x^9}{9} \]

[In]

int(x^2*(a + b*x^2)^2*(c + d*x^2),x)

[Out]

x^5*((a^2*d)/5 + (2*a*b*c)/5) + x^7*((b^2*c)/7 + (2*a*b*d)/7) + (a^2*c*x^3)/3 + (b^2*d*x^9)/9